观测器误差动力学线性化的简单验证与实现：帕斯卡三角-海森矩阵准则
Simple Verification and Implementation of Observer Error Dynamics Linearization: A Pascal's Triangle--Hessian Matrix Criterion
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中文总结 AI 辅助
本文针对单输出系统提出帕斯卡-海森条件，将观测器误差线性化充要条件转化为海森矩阵结构检验，降低计算复杂度，还推广至多输出系统，数值例子验证了方法有效性。
中文摘要 AI 辅助
自20世纪80年代创立以来，非线性观测器误差线性化的经典理论——非线性观测器典范形式——一直受到持续关注。在现有理论框架下，验证与构造非线性系统实现观测器误差线性化虽有系统高效的实现方式，但严重限制了该理论在高维系统中的适用性。为解决这一问题，受高阶全量测系统定义的启发，本文针对单输出系统提出帕斯卡-海森条件。该条件将观测器误差线性化的充要条件等价转化为对高阶全量测系统中非线性项海森矩阵的结构检验：海森矩阵左上角的系数构成帕斯卡三角，右下角则恒为零。同时，本文为典范形式中所有依赖输出的单变量函数提供了显式积分公式，无需求解偏微分方程。与现有理论相比，本文方法将条件验证的计算复杂度从$O(n^4)$降至$O(n^2)$，并以构造典范形式的显式不定积分替代了复杂的偏微分方程求解过程。本文还将结果推广到具有相同可观测性指数的多输出系统。除计算优势外，该研究揭示了非线性观测器典范形式与帕斯卡三角之间自20世纪80年代理论创立以来未被注意到的基本结构联系。数值例子验证了所提方法的有效性。
英文摘要
The classical theory of nonlinear observer error linearization---the nonlinear observer canonical form---has attracted sustained attention since its inception in the 1980s. Under the existing theoretical framework, the verification and construction for a nonlinear system to achieve observer error linearization admits a systematic, efficient implementation, severely limiting the applicability of the theory to high-dimensional systems. To address this issue, inspired by the definition of high-order fully measured systems, this paper proposes the Pascal-Hessian condition for single-output systems. This condition equivalently converts the necessary and sufficient condition for observer error linearization into a structural test on the Hessian matrix of the nonlinear term in high-order fully measured systems: the coefficients in the upper-left corner of Hessian matrix form a Pascal's triangle, while the lower-right corner vanishes identically. Simultaneously, we provide explicit integral formulas for all output-dependent univariate functions in the canonical form, eliminating the need to solve partial differential equations. Compared with the existing theory, our method reduces the computational complexity of condition verification from $O(n^4)$ to $O(n^2)$, and replaces the intricate process of solving partial differential equations with explicit indefinite integral for canonical form construction. We further extend the result to multi-output systems with equal observability indices. Beyond its computational advantages, this work reveals a fundamental structural connection between the nonlinear observer canonical form and Pascal's triangle---a link that has remained unnoticed since the inception of the theory in the 1980s. Numerical examples validate the effectiveness of the proposed method.
